A tropical curve is a vertex-weighted metric graph. It is hyperelliptic if it admits an involution whose quotient is a tree. Assuming no prior knowledge of tropical geometry, I will develop the theory of tropical hyperelliptic curves and discuss the relationship with classical algebraic curves and their Berkovich skeletons. Along the way, we will see some nice combinatorics, including an analogue for graphs of holomorphic maps of Riemann surfaces.